Abstract
In the paper we focus on self-adjoint noncommutative martingales. We provide an extension of the notion of differential subordination, which is due to Burkholder in the commutative case. Then we show that there is a noncommutative analogue of the Burkholder method of proving martingale inequalities, which allows us to establish the weak type (1,1) inequality for differentially subordinated martingales. Moreover, a related sharp maximal weak type (1,1) inequality is proved.
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Burkholder D.L. (1966). Martingale transforms. Ann. Math. Statist. 37: 1494–1504
Burkholder, D.L.: Sharp inequalities formartingales and stochastic integrals.Astésque , 75–94 (1988)
Fack T., Kosaki H. (1986). Generalized s-numbers of τ-measurable operators. Pacific J. Math. 123(2): 269–300
Junge M. (2002). Doob’s inequality for noncommutative martingales. J. Reine Angew. Math 549: 149–190
Junge M., Musat M. (2007). A noncommutative version of the John–Nirenberg Theorem. Trans. Am. Math. Soc. 359(1): 115–142
Junge M., Xu Q. (2003). Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31(2): 948–995
Musat M. (2003). Interpolation between noncommutative BMO and noncommutative L p spaces. J. Funct. Anal. 202: 195–225
Parcet, J., Randrianantoanina, N.: Gundy’s decomposition for noncommutative martingales and applications. Proc. Lond. Math. Soc. (3) 93(1), 227–252 (2006)
Pisier G., Xu Q. (1997). Noncommutative martingale inequalities. Commun. Math. Phys. 189: 667–698
Pisier, G., Xu, Q.: Noncommutative L p -spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces II. North Holland, Amsterdam, pp. 1459–1517 (2003)
Randrianantoanina N. (2002). Noncommutative martingale transforms. J. Funct. Anal. 194: 181–212
Randianantoanina N. (2004). Square function inequalities for noncommutative martingales. Israel J. Math. 140: 333–365
Randrianantoanina N. (2005). A weak-type inequality for noncommutative martingales and applications. Proc. Lond. Math. Soc. 91(3): 509–544
Randrianantoanina, N.: Conditioned square functions for noncommutative martingales, Preprint
Takesaki M. (1979). Theory of operator algebras I. Springer, New York
Xu, Q.: Recent development on noncommutative martingale inequalities. Functional Space Theory and its Applications. In: Proceedings of International Conference & 13th Academic Symposium in China, Wuhan, Research Information Ltd, UK, pp. 283–314 (2003)
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Research supported by MEN Grant 1 PO3A 012 29.
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Osȩkowski, A. Weak type inequality for noncommutative differentially subordinated martingales. Probab. Theory Relat. Fields 140, 553–568 (2008). https://doi.org/10.1007/s00440-007-0075-0
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DOI: https://doi.org/10.1007/s00440-007-0075-0
Keywords
- Noncommutative probability space
- Martingale
- Weak type (1,1) inequality
- Differentially subordinated martingales